In the paper we consider the macroscopic model of plasma of scalar charged particles, obtained by means of the statistical averaging of the microscopic equations of particle dynamics in a scalar field. On the basis of kinetic equations, obtained from averaging, and their strict integral consequences, a self-consistent set of equations is formulated which describes the self-gravitating plasma of scalar charged particles. It was obtained the corresponding closed cosmological model which also was numerically simulated for the case of one-component degenerated Fermi gas and two-component Boltzmann system. It was shown that results depend weakly on the choice of a statistical model. Two specific features of cosmological evolution of a statistical system of scalar charged particles were obtained with respect to cosmological evolution of the minimal interaction models: appearance of giant bursts of invariant cosmological acceleration $\Omega$ at the time interval $8\cdot10^3\div2\cdot10^4 t_{Pl}$ and strong heating ($3\div 8$ orders of magnitude) of a statistical system at the same times. The presence of such features can modify the quantum theory of generation of cosmological gravitational perturbations.

The number of superheavy particles with the mass of the Grand Unification scale with trans-Planckian energy created at the epoch of superheavy particle creation from vacuum by the gravitation of the expanding Universe is calculated. In later collisions of these particles gravitational radiation is radiated playing the role of bremsstrahlung for gravity. The effective background radiation of the Universe is evaluated.